It seems the WC would be better on the top wall with the vanity. If you have a walk-in shower, it doesn't need to be quite so wide, and you can move the entrance to the bottom to be more generous. If you have a tub/shower combo, you don't need the partition at all. Consider flipping the door swing unless you're really concerned with privacy.

You can install an underfloor heating system which is a series of pipes running under the floor where hot water runs and creates a warm floor and also heats the room with radiant heat. This is actually the most effective form of heating a space and is the most comfortable. The system components can be purchased or you can have a licensed contractor install it for you. You can put just about any type of flooring over the top......tile, laminate wood,........ Regarding flooring, if it is a very wet bathroom, I would prefer tile, but if will not get a lot of moisture you can use wood or laminate flooring.

Based on the dimensions you have given and the religious aspects you mention you are very restricted in what the layout can be. The best solution I see means you have to move the door towards the south by 36". I suggest you make it a pocket door retracting towards the south wall. This leaves 36" x 60" of space on the north wall for either a shower or tub. The vanity would be on the west wall ( across from the door) spaced away from the tub or shower around 12". Make sure the vanity cabinet is shallow, no more than 21" deep. The vanity should be able to be about 48" wide. The toilet will go on the South wall with the right side of it close to the East wall. Code calls for approx. 33" of width for the toilet overall. If budget allows you could continue the vanity cabinets into the corner of the south wall however keep in mind that most of that area that will be next to the toilet and not easily accessible. But it would provide extra storage or counter space. The sink will have to be centered in the 48" part of the vanity. You could also frame out the southwest corner to capture the side of the vanity instead of a longer vanity. Hope this helps.

You will have to reroute the sewer pipe into the wall and cap off the existing. You have to make sure you have a clear bay where the toilet goes and there is no obstructions, sometimes framing has to be re-done so you have to open up a wall etc and have to patch the adjoining room behind it as well. Call someone local to come in and evaluate the situation and they will tell you what you need based on their evaluation. Good luck.